Controlled charging of battery electric vehicles (BEVs) is intensely discussed and a focus of current research. The temporal and spatial flexibility of the electricity demand stemming from battery electric vehicles can be used advantageously for the power system.
However, also battery ageing effects occur during charging of a BEV. Hence, the costs for charging BEVs consist of electricity cost as well as battery ageing cost. Due to the substantial share of the battery cost in the total cost of a BEV, it is crucial to consider the battery ageing effects during the charging process besides electricity price or other power system restrictions.
This thesis presented a mathematical optimisation model, proposing optimal charging strategies for BEVs while taking battery ageing behaviour as well as fluctuating electricity prices into account. While considering vehicle parameters and travel requirements of the BEV users, charging profiles at minimal cost were generated. The aim of the charging optimisation model was to find the optimal trade-off between electricity price induced charging and saving battery lifetime.
In order to simulate the travel requirements of the BEVs, which shall be optimised in the charging optimisation, an agent-based mobility model was developed, representing the driving and parking behaviour in Singapore. The model was based on statistical data on travel behaviour in Singapore. It comprises a vehicle model for calculation of energy consumption of BEVs, based on vehicle parameters of currently available BEV models. A BEV population corresponding to the current vehicle population in Singapore was generated. Travel schedules and corresponding energy consumptions of a sample of BEVs were simulated over a specific time period and served as input for the calculations of the charging optimisation model.
To model the battery ageing behaviour taking place during the charging process of a BEV, battery ageing tests were conducted. Both cycle and calendar ageing tests were performed in order to analyse the ageing effects during operation and storage of a battery. The physical quantity of energy content was chosen to measure the battery ageing effects, including both capacity fade and impedance rise.
During the cycle ageing tests, the test cells were cycled between different states of charge (SOCs) and with different charge rates. Thereby, different influence factors were examined. In some of the tests, different charge rates were applied, while the SOC range
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was kept constant. In other cycle ageing tests, the charge rate remained unchanged and the SOC range in which the cells were cycled was varied. The development of the energy fade depending on the different influence factors was fitted with two different functions:
one exponential function of the charge rate as well as another exponential function depending on the SOCstartand SOCendof the charging process. The combination of these two fitted functions resulted in a multidimensional cycle ageing function, indicating the energy fade for any combination of SOCstart, SOCend, and charge rate. The analysis of the cycle ageing tests showed that higher charge rates as well as charging to high SOCs or discharging to low SOCs accelerates cycle ageing.
For the calendar ageing tests, cells were stored at different temperatures and different SOCs and measured after regular time intervals. The energy fade during storage at different SOCs was approximated by two linear equations. The calendar ageing tests yielded higher energy fade for higher SOCs.
Both the multidimensional cycle ageing function as well as the two linear functions ap-proximating the calendar ageing behaviour were integrated in the charging optimisation model. The ageing functions provided the basis for the calculation of the battery ageing costs (divided into cycle and calendar ageing costs) as part of the objective function.
The objective function minimised the total charging costs, consisting of electricity cost and ageing cost. To reach this objective, the optimisation model allocated the charging processes by deciding how the BEVs were recharged: during which parking event and specifically during which time steps thereof, from what SOCstartto what SOCendand at what charge rate. Thereby, the travel requirements of the BEVs had to be met: The BEVs always had to be able to fulfil the subsequent trips and they were not allowed to violate a minimum SOC limit representing a buffer. The charging optimisation model yielded charging strategies for BEVs which made use of periods with low electricity prices and preserved the batteries’ lifetime at the same time.
The optimisation problem was initially formulated as a mixed-integer non-linear pro-gramme. By applying the reformulation-linearisation technique to the charge power term, the non-linear constraints containing this term were reformulated as linear constraints.
Also the non-linear, convex cycle ageing function was piecewise-linearly approximated by tangent hyperplanes. Thereby, the mathematical formulation of the optimisation problem could be transformed into a mixed-integer linear programme. The linearisation led to increased performance during solving the optimisation problem.
In case of non-convex battery ageing data, an alternative solution for the piecewise linear approximation of the multidimensional cycle ageing function was elaborated.
A theoretical approach of multidimensional interpolation by means of SOS2 variables was derived and included in the charging optimisation problem. Thus, the charging optimisation model was made available for a general application with any cell chemistry which might feature non-convex ageing behaviour.
The developed charging optimisation model was used to find optimal charging strategies for a sample of 300 BEVs. The travel schedules and corresponding energy consumption of the BEVs were generated by the mobility model. The optimisation horizon was set to one week and different scenarios were analysed. The scenarios included a base scenario, a scenario with future battery cost development, a scenario
113 with future battery performance, as well as a scenario with a high charge rate.
The battery ageing costs resulted in a share of 13 % to 45 % of the total charging costs, depending on the scenario. Even for the most optimistic scenario with 13 % (future battery cost scenario), this is a substantial share which should not be omitted when optimising charging strategies. Especially because the cost of battery ageing will rise even more when not considered in the charging optimisation model. Also, the batteries of BEVs would reach their end of life much earlier if battery ageing is omitted and only power system related aspects are optimised.
The results of the scenario analysis showed that the BEVs should be operated within a SOC range of 10 % to 50 % to keep calendar ageing and also cycle ageing effects at a minimum. Substantial damage to the battery due to deep discharge below 10 % is avoided by the minimum SOC limit. Fast charging was applied only very seldomly due to its considerable influence on battery ageing and mostly low charge rates, resulting in charging times of 2.5 h to 5 h (relating to a full charge), were applied. Thereby, the cycle ageing cost remained at a low level. Even for the scenario with future battery performance, in which fast charging had a less severe effect on battery ageing, the possible savings in electricity cost due to charging more energy in low-priced periods could not outweigh the increased battery ageing cost. An improved ageing behaviour at higher charge rates does not have an effect on the charging strategies, unless this improvement is far superior than assumed in the scenario analysis conducted in this work. The effect of fast charging became obvious in the scenario in which the BEVs were charged at a charge rate of 1 P (equalling a full charge in 1 h). In this scenario, the battery ageing cost rose to a value more than three times as high as in the base scenario and reached a share of 45 % in the total charging costs.
Another outcome of the calculations of the charging optimisation model was that most BEVs were charged in the early morning hours when electricity prices and also the electricity demand in Singapore were at their minima. The BEVs were charged when electricity price and demand were still low, but as late as possible before they departed in order to keep the average SOC and therewith calendar ageing low. Other points of time used for charging coincided with smaller valleys in electricity price and demand or periods with relatively low values of the aforementioned. As a consequence, charging electricity cost were kept at a low level and valley-filling effects occurred in the power system.
When a very large amount of BEVs shall be optimised in terms of their charging processes, the additional constraint on the power system load, implemented in the charging optimisation model, will take effect. Additional load peaks due to excessive BEV charging at the same time can be avoided and the charging processes are distributed over a longer time window. The power system load constraint can also be applied to single planning areas, thereby implementing different load restrictions for different areas of Singapore and representing, for example, possible local bottlenecks at substations or the distribution grid. The mobility model already assigned the trips and parking locations to the different planning areas of Singapore during simulation of the travel schedules. Another effect of a large number of BEVs charging at the same time can be rising prices during these periods and, as a consequence thereof, a shift in charging load
to other periods.
The charging optimisation model can be expanded to include further ageing pa-rameters as, for example, varying battery temperature. To integrate additional ageing parameters, the battery ageing function can be extended by another dimension reflecting the influence of an extra parameter. Both the approach of piecewise linear approximation by tangential hyperplanes (for convex ageing functions) and by means of SOS2 vari-ables (for non-convex ageing functions) is applicable to ageing functions of even more dimensions. The tangential hyperplanes then would have as many dimensions as ageing parameters. Also the dimensions and quantity of SOS2 variables and corresponding constraints are scaled up according to the number of ageing parameters.
Another aspect to be analysed is the optimal battery dimensioning in terms of energy content in order to fulfil the travel requirements of BEV users. Larger batteries have additional weight, resulting in a higher energy consumption of the BEVs. This effect was already incorporated in the mobility model. However, smaller batteries need to be operated closer to their limits in order to still fulfil all travel requirements and thus experience increased ageing. Hence, in future research, a coupling of the charging optimisation with an optimisation for battery dimensioning makes sense to find the optimal trade-off between reducing battery cost due to smaller batteries and increasing ageing effects because of more extreme battery operation.
Even though the charging optimisation model indicated that it is best to operate one’s BEV always close to the lower SOC limit, many users might not feel comfortable by doing so and privilege their personal preferences over the optimal charging patterns in terms of battery ageing and electricity cost. To reflect this, user preferences can be embraced in the definition of the agents’ travel behaviour in the mobility model.
Instead of optimising the charging processes for a fixed time horizon, whereat one cannot be totally sure about the exact travel schedules of the agents, the approach of a receding horizon can be used. The Energy Market Company (2016b) provides a forecast of the electricity prices for 24 h to 36 h, which is updated every 30 min. With this forecast and the current prognosis of the travel schedules, the next 24 h can be optimised. After 30 min, the following 24 h are optimised, including updated electricity prices and travel schedules. Thereby, short-term changes in travel schedules can be considered and a real-time application of the charging optimisation model becomes possible. In contrast to the simulation horizon of one week, the flexibility to shift charging processes among different parking events decreases due to the shorter optimisation horizon. However, computation times will shorten which is important for real-time applications. The charging optimisation can take place separately for each vehicle and the intelligence can be located within each vehicle. However, when a very large amount of BEVs shall be optimised and power system restrictions come into play, the charging load of all optimised BEVs needs to be coordinated and distributed over space and time. Therefore, all BEVs have to be connected and the control will take place within this network or cloud.
Another possible application of the battery ageing model and the battery ageing constraints derived therefrom is within a load aggregator, which combines flexible loads of BEVs to take part in a demand response programme as it was done by Recalde Melo,
115 Trippe, Gooi, and Massier (2016).
Instead of optimising the charging strategies for traction batteries in BEVs, the optimisation model could also be applied to stationary battery storage systems for the power grid. Replacing the input generated by the mobility model, which represents the BEVs’ energy requirements resulting from their travel schedules, with profiles of power system load and renewable energy production, optimal charge and discharge profiles for the stationary batteries can be elaborated.
Summing up, the battery ageing costs occurring during charging processes which were analysed and determined within this work exhibit the necessity to include battery ageing behaviour into a charging optimisation model. The way of charging a BEV has an immense influence on the lifetime of the battery. Recommended charging strategies regarding only power system aspects should be reconsidered. When omitting battery ageing effects during the optimisation of charging strategies, battery ageing costs will rise and BEVs will reach their end of life sooner. Slow charge rates and operation in low to moderate SOC ranges proved to be advantageous charging strategies. Charging processes were predominantly scheduled during periods of low electricity prices, coinciding with valleys or low levels of electricity demand. The benefit of the charging optimisation model elaborated in this work is that also battery ageing aspects are considered and formulated as part of a comprehensive mathematical optimisation model for charging strategies of battery electric vehicles.